https://file.aiquickdraw.com/mathbot/file/1730068992714zy99iowz.jpg

The problem asks for a redrawing of each graph to eliminate edge crossings while keeping them planar. Let’s examine how each of these graphs can be redrawn:

1. **First Graph**: The first graph shows a closed shape with intersecting lines in the center. This structure resembles a "bowtie" or "butterfly" graph. To make it planar, try moving one vertex outside of the current plane, which can eliminate the central crossing. Alternatively, you could split the central crossing by re-routing one edge around the outside of the shape.

2. **Second Graph**: The second graph appears to be a bipartite graph, possibly resembling \( K_{3,3} \) (complete bipartite graph). To draw this graph without edge crossings, consider re-arranging the vertices into two rows, with three vertices on each row, and draw curved edges connecting vertices without intersecting each other. 

3. **Third Graph**: The third graph resembles a cube or a square grid with diagonals. Redrawing this without intersections involves adjusting the layout to remove diagonal crossings. For instance, you could place vertices in a 3D arrangement (like in a cube) or use curved edges.

Let me know if you would like a more specific step-by-step redrawing of each graph. I can also provide a sketch or additional details to help visualize these planar configurations.

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Would you like further details or sketches? Here are some related questions:

1. What is a planar graph, and how can you determine if a graph is planar?
2. What is the importance of the Kuratowski’s theorem in graph planarity?
3. How does the concept of graph embedding apply to solving these types of problems?
4. Are there algorithms to test planarity in graphs automatically?
5. How can you verify if your redrawn graph has no edge crossings?

**Tip:** When redrawing graphs to eliminate edge crossings, try to visualize curves or bends in the edges, as this often helps to separate intersecting parts.

Don't cross in the edge. Each of the graphs drawn below is actually planar. Redraw each one so that none of the edges cross.

Learn how to redraw graphs to ensure they remain planar without edge crossings. This problem introduces key concepts in graph theory, including Kuratowski's Theorem, and provides methods for redrawing intersecting graphs to avoid overlapping edges.

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